This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.

In a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n\ge 1$, this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system

\begin{eqnarray*} \left \{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[5pt] v_t = \Delta v + u - \alpha uv, \end{array} \right . \end{eqnarray*}

$\alpha \gt 0$

It is shown that there exists $\delta _\star =\delta _\star (n)\gt 0$ such that for any given $\alpha \ge \frac{1}{\delta _\star }$ and for any suitably regular initial data satisfying $v(\cdot, 0)\le \delta _\star$, this problem admits a unique classical solution that stabilizes to the constant equilibrium $(\frac{1}{|\Omega |}\int _\Omega u(\cdot, 0), \, \frac{1}{\alpha })$ in the large time limit.

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant

\[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \]

$\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$

$\partial \Omega$

$m>\max \{1,\,\frac {3N-2}{2N+2}\}$

$(\bar {u}_0,\,0)$

$t\rightarrow \infty$

$\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$

$m>\frac {3N-2}{2N}$

$N\geq 3$

$m>\frac {3N}{2N+2}$

$N\geq 2$

This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$. The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.

We consider the global existence for the following fully parabolic chemotaxis system with two populations

\[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, & x\in\Omega,\ t>0, \\ v_t=\Delta v-v+u_1+u_2, & x\in\Omega,\ t>0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), & x\in\Omega, \end{array} \right. \]

$\Omega =\mathbb {R}^2$

$\Omega =B_R(0)\subset \mathbb {R}^2$

$\kappa _i,\chi _i>0,$

$i=1,2$

$0=\Delta v-v+u_1+u_2$

$0=\Delta v+u_1+u_2$

In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$, the chemotaxis system

\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]

$u$

$v$

$D\in C^{3}([0,\infty ))$

$0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$

$\xi >0$

${K_D}>0$

$\alpha >0$

$\Omega$

We study stationary solutions to the Keller–Segel equation on curved planes. We prove the necessity of the mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case. Furthermore, we provide a correspondence between stationary solutions to the static Keller–Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller–Segel equation (with any mass). Finally, as a complementary result, we prove a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality and use it to show that the Keller–Segel free energy is bounded from below exactly when the mass is $8 \pi$, even in the curved case.

Host-seeking behaviour and how a parasite identifies the correct host to infect remains a poorly understood area of parasitology. What is currently known is that host sensation and seeking behaviour is formed from a complex mixture of chemo-, thermo- and mechanosensory behaviours, of which chemosensation is the best studied. Previous studies of olfaction in parasitic nematodes suggested that this behaviour appears to be more closely related to target host and infection mode than phylogeny. However, there has not yet been a study comparing the chemotactic and temperature-dependent behaviours of very closely related parasitic and non-parasitic nematodes. To this end, we examined the temperature-dependent and chemotactic responses of the Strongyloidoidea superfamily of nematodes. We found differences in temperature response between the different species and within infective larvae. Chemotactic responses were highly divergent, with different attraction profiles between all species studied. When examining direct stimulation with fur, we found that it was insufficient to cause an attractive response. Overall, our results support the notion that olfactory sensation is more closely related to lifestyle and host range than phylogeny, and that multiple cues are required to initiate host-seeking behaviour.

The gastropod parasitic nematode Phasmarhabditis hermaphrodita has been formulated into a successful biological control agent (Nemaslug®, strain DMG0001) used to kill slugs on farms and gardens. When applied to soil, P. hermaphrodita uses slug mucus and faeces to find potential hosts. However, there is little information on what cues other species of Phasmarhabditis (P. neopapillosa and P. californica) use to find hosts and whether there is natural variation in their ability to chemotax to host cues. Therefore, using chemotaxis assays, we exposed nine wild isolates of P. hermaphrodita, five isolates of P. neopapillosa and three isolates of P. californica to mucus from the pestiferous slug host Deroceras invadens, as well as 1% and 5% hyaluronic acid – a component of slug mucus that is highly attractive to these nematodes. We found P. hermaphrodita (DMG0010) and P. californica (DMG0018) responded significantly more to D. invadens mucus and 1% hyaluronic acid than other strains. Also, P. hermaphrodita (DMG0007), P. neopapillosa (DMG0015) and P. californica (DMG0017) were superior at locating 5% hyaluronic acid compared to other isolates of the same genera. Ultimately, there is natural variation in chemoattraction in Phasmarhabditis nematodes, with some strains responding significantly better to host cues than others.

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T > 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

This paper deals with the logistic Keller–Segel model

\[ \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu u^2, \\ v_t = \Delta v - v + u \end{cases} \]

$\bar {\Omega }\times (0,\infty )$

We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Carbon dioxide (CO2) is an important sensory cue for many animals, including both parasitic and free-living nematodes. Many nematodes show context-dependent, experience-dependent and/or life-stage-dependent behavioural responses to CO2, suggesting that CO2 plays crucial roles throughout the nematode life cycle in multiple ethological contexts. Nematodes also show a wide range of physiological responses to CO2. Here, we review the diverse responses of parasitic and free-living nematodes to CO2. We also discuss the molecular, cellular and neural circuit mechanisms that mediate CO2 detection in nematodes, and that drive context-dependent and experience-dependent responses of nematodes to CO2.

Male gamete chemotaxis towards the female gamete is a general strategy to facilitate the sexual reproduction in many marine eukaryotes. Biochemical studies of chemoattractants for male gametes of brown algae have advanced in the 1970s and 1980s, but the molecular mechanism of male gamete responses to the attractants remains elusive. In sea urchin, a K+ channel called the tetraKCNG channel plays a fundamental role in sperm chemotaxis and inhibition of K+ efflux through this channel by high K+ seawater blocks almost all cell responses to the chemoattractant. This signalling mechanism could be conserved in marine invertebrates as tetraKCNG channels are conserved in the marine invertebrates that exhibit sperm chemotaxis. We confirmed that high K+ seawater also inhibited sperm chemotaxis in ascidian, Ciona intestinalis (robusta), in this study. Conversely, the male gamete chemotaxis towards the female gamete of a brown alga, Mutimo cylindricus, was preserved even in high K+ seawater. This result indicates that none of the K+ channels is essential for male gamete chemotaxis in the brown alga, suggesting that the signalling mechanism for chemotaxis in this brown alga is quite different from that of marine invertebrates. Correlated to this result, we revealed that the channels previously proposed as homologues of tetraKCNG in brown algae have a distinct domain composition from that of the tetraKCNG. Namely, one of them possesses two repeats of the six transmembrane segments (diKCNG) instead of four. The structural analysis suggests that diKCNG is a cyclic nucleotide-modulated and/or voltage-gated K+ channel.

Phasmarhabditis hermaphrodita is a parasitic nematode of terrestrial gastropods that has been formulated into a biological control agent for farmers and gardeners to kill slugs and snails. In order to locate slugs it is attracted to mucus, faeces and volatile cues; however, there is no information about whether these nematodes are attracted to snail cues. It is also unknown how wild isolates of P. hermaphrodita or different Phasmarhabditis species behave when exposed to gastropod cues. Therefore, we investigated whether P. hermaphrodita (commercial and wild isolated strains), P. neopapillosa and P. californica were attracted to mucus from several common snail species (Cepaea nemoralis, Cepaea hortensis, Arianta arbustorum and Cornu aspersum). We also examined whether snails (C. aspersum) collected from different locations around the UK differed in their attractiveness to wild isolates of P. hermaphrodita. Furthermore, we also investigated what properties of snail mucus the nematodes were attracted to, including hyaluronic acid and metal salts (FeSO4, ZnSO4, CuSO4 and MgSO4). We found that the commercial strain of P. hermaphrodita responded poorly to snail mucus compared to wild isolated strains, and C. aspersum collected from different parts of the UK differed in their attractiveness to the nematodes. We found that Phasmarhabditis nematodes were weakly attracted to all metals tested but were strongly attracted to hyaluronic acid. In a final experiment we also showed that pharmacological manipulation of cyclic guanosine monophosphate (cGMP) increased chemoattraction to snail mucus, suggesting that the protein kinase EGL-4 may be responsible for Phasmarhabditis sp. chemoattraction.

The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given by

under Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.

We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.